Michael Carcone and Travis Boyd also scored for Arizona (22-42-5), which had lost seven of eight. Karel Vejmelka made 29 saves after he was pulled in the first period of a 5-0 loss to the Anaheim Ducks on Friday.
"That was huge the way we played," Coyotes coach Andre Tourigny said. "The win for sure will kind of cap off everything, but the way we played, we were way more in sync. We talked a lot this morning about having more cohesion, being more in sync. Offensively, we were much better, more cohesion, more on the same page."
Dylan Strome and Patrick Kane scored, and Lankinen made 32 saves for Chicago (24-35-11), which has lost five in a row (0-3-2).
"Yeah, every day feels like Groundhog Day right now," Blackhawks coach Derek King said. "It was kind of a funny game. We come out, we started well and the second period, I don't know if we're tired from all this travel we've been doing or the busy schedule, but we just stop thinking, shut our brains off and let them get back in the game and take the lead.
"I did like our response in the third. We played well. I thought [Lankinen] played a great game. He was awesome. He deserved a better outcome the way he played. We just got a little too sloppy and a little too high risk in front of him there at the end. We just can't do that."
Prior to the game, the Blackhawks honored Toews, their captain, for playing in his 1,000th NHL game against the Florida Panthers on Thursday.
"Just in a minute here I'll cool down after what just happened this game and definitely get back to appreciating how special it was," Toews said. "I was pretty humbled and pretty touched with the recognition from my teammates."